The aim of this work is to use the unification of finite element methods (FEM) and computer aided design (CAD) embedded in isogeometric analysis to solve shape optimization problems within fluid mechanics. The flow problems considered are governed by the 2-dimensional steady-state, incompressible Navier-Stokes equations. These partial differential equations are solved for fluid velocity and pressure using B-spline based isogeometric analysis. The accurate geometry representation and high degree of continuity of the flow fields are some of the method's advantages. To ensure stable discretizations, though, care has to taken in the choice of polynomial degrees and knots vectors for the velocity and pressure approximations. In shape optimization for fluids we search for an optimal design of the flow domain that minimizes a prescribed objective, while satisfying suitable constraints. With the ability to represent complex shapes in few design variables, and the unification of the analysis and geometry models, isogeometric analysis is highly suited for shape optimization purposes. The design variables are the coordinates of control points that define the boundary of the domain. As the optimizer moves the control points around, though, control points are sometimes seen to coallesce and the control net might even fold over severely, causing an improper design and the analysis to break down. Regularization methods to ensure a good boundary representation are therefore often needed. The methodology is presented through a simple example in which a pipe bend is designed to minimize the drag with a constraint on the area of the pipe. The basics of the analysis of the Navier-Stokes equations are briefly covered, some regularization methods to ensure good boundary parametrisations during optimization are discussed, and different design results for a range of Reynolds numbers are presented.
Place: Austin, Texas, USA

The aim of this work is to use the unification of finite element methods (FEM) and computer aided design (CAD) embedded in isogeometric analysis to solve shape optimization problems within fluid mechanics. The flow problems considered are governed by the 2-dimensional steady-state, incompressible Navier-Stokes equations. These partial differential equations are solved for fluid velocity and pressure using B-spline based isogeometric analysis. The accurate geometry representation and high degree of continuity of the flow fields are some of the method's advantages. To ensure stable discretizations, though, care has to taken in the choice of polynomial degrees and knots vectors for the velocity and pressure approximations. In shape optimization for fluids we search for an optimal design of the flow domain that minimizes a prescribed objective, while satisfying suitable constraints. With the ability to represent complex shapes in few design variables, and the unification of the analysis and geometry models, isogeometric analysis is highly suited for shape optimization purposes. The design variables are the coordinates of control points that define the boundary of the domain. As the optimizer moves the control points around, though, control points are sometimes seen to coallesce and the control net might even fold over severely, causing an improper design and the analysis to break down. Regularization methods to ensure a good boundary representation are therefore often needed. The methodology is presented through a simple example in which a pipe bend is designed to minimize the drag with a constraint on the area of the pipe. The basics of the analysis of the Navier-Stokes equations are briefly covered, some regularization methods to ensure good boundary parametrisations during optimization are discussed, and different design results for a range of Reynolds numbers are presented.
Place: Austin, Texas, USA